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  <h1 class="w3-xxxlarge">Group Explorer 3.0 Help: Group Theory Terminology</h1>
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                <h3>Contents</h3>
                <ul>
                
                    <li><a style="text-decoration: none;"
                          href="#1-1-one-to-one">1-1 ("one-to-one")</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#abelian-group">Abelian group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#bijection-bijective">Bijection, bijective</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cayley-diagrams">Cayley diagrams</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cayley-table">Cayley table</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#class-equation">Class equation</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#closure-of-a-subset">Closure of a subset</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#commutative-group">Commutative group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#conjugacy-classes">Conjugacy classes</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cosets">Cosets</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cycle-graph">Cycle graph</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cyclic-group">Cyclic group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#definition-of-a-group-via-generators-and-relations">Definition of a group via generators and relations</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#elementwise-product-of-two-subsets-of-a-group">Elementwise product (of two subsets of a group)</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#epimorphism">Epimorphism</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#first-isomorphism-theorem">First Isomorphism Theorem</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#generators-for-a-group-or-subgroup">Generators for a group (or subgroup)</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#group">Group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#homomorphism">Homomorphism</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#image-of-a-subset-under-a-morphism">Image (of a subset under a morphism)</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#index-of-a-subgroup">Index of a subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#injective-injection">Injective, injection</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#isomorphism-isomorphic">Isomorphism, isomorphic</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#the-isomorphism-theorems">The isomorphism theorems</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#kernel-of-a-homomorphism">Kernel (of a homomorphism)</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#lattice-of-subgroups">Lattice of subgroups</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#left-cosets">Left cosets</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#monomorphism">Monomorphism</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#morphism">Morphism</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#multiplication-table">Multiplication table</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#normal-subgroup">Normal subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#normalizer-of-a-subgroupsubset">Normalizer of a subgroup/subset</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#objects-of-symmetry">Objects of symmetry</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#one-to-one">One-to-one</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#onto">Onto</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#orbit-of-an-element-in-a-group">Orbit of an element in a group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#order-classes">Order classes</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#order-of-an-element-in-a-group">Order of an element in a group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#order-of-a-group">Order of a group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#order-of-a-subgroup">Order of a subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#p45subgroup">p-subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#preimage-of-a-subset-under-a-morphism">Preimage (of a subset under a morphism)</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#proper-subgroup">Proper subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#quotient-group">Quotient group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#right-cosets">Right cosets</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#short-exact-sequence">Short exact sequence</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#simple-group">Simple group</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#solvable-group-solvable-decomposition">Solvable group, solvable decomposition</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#subgroup">Subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#surjective-surjection">Surjective, surjection</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#sylow-p-subgroup">Sylow p-subgroup</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#symmetry-objects">Symmetry objects</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#trivial-groupsubgroup">Trivial group/subgroup</a></li>
                
                </ul>

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        <div class="w3-threequarter w3-container">
            <p>This page is a dictionary of many of the essential terms one might come
across when beginning to learn group theory. Many other <em>Group Explorer</em>
help pages link here to define terms. Unlike <a href="../rf-geterms/">the <em>Group Explorer</em>
terminology page</a>, these terms not specific to <em>Group
Explorer itself</em>; all are all commonly used mathematical terms.</p>
<h3 id="1-1-one-to-one">1-1 (&ldquo;one-to-one&rdquo;)</h3>
<p>See <a href="#injective-injection">injective</a>.</p>
<h3 id="abelian-group">Abelian group</h3>
<p>An abelian group is one whose binary operation is commutative. That is, for
every two elements <script type="math/tex">a</script> and <script type="math/tex">b</script> in the group, <script type="math/tex">a\cdot b=b\cdot a</script>.</p>
<p>CITE(VGT-5.2 MM-2.1 TJ-13.1)</p>
<h3 id="bijection-bijective">Bijection, bijective</h3>
<p>A function that is both <a href="#injective-injection">injective</a> and
<a href="#surjective-surjection">surjective</a> is called bijective.</p>
<h3 id="cayley-diagrams">Cayley diagrams</h3>
<p>A Cayley diagram is a graph (that is, a set of vertices and edges among
them) that depicts a group. There is one node (vertex) in the graph for each
element in the group, and the arrows (edges) show how the generators act on
the elements of the group.</p>
<p>For instance, if the group has two generators, <script type="math/tex">a</script> and <script type="math/tex">b</script>, then there
will be one type of arrow (perhaps red-colored arrows) for generator <script type="math/tex">a</script>
and another type of arrow (perhaps blue-colored) for generator <script type="math/tex">b</script>. You
can see that the following Cayley diagram fits this description.</p>
<p><img alt="Cayley diagram of S_3" src="../s_3_cayley_miniature.png" /></p>
<p>The red arrows connect two elements if multiplying the first by <script type="math/tex">a</script> gives
the second. That is, we have <script type="math/tex">x~{\color{red}\longrightarrow}~y</script> just when
<script type="math/tex">x\cdot a=y</script>. So the arrows represent right-multiplication. (One could
also make Cayley diagrams in which the arrows represented
left-multiplication.)</p>
<p><em>Group Explorer</em> has a <a href="../rf-geterms/#visualizers">visualizer</a> for showing
Cayley diagrams. It is documented in full <a href="../rf-um-cd-options/">here</a>,
with an introduction <a href="../gs-cd-intro/">here</a>, and a tutorial
<a href="../tu-cd-manip/">here</a>.</p>
<p>CITE(VGT-2.4 MM-1.2 DE-3)</p>
<h3 id="cayley-table">Cayley table</h3>
<p>See <a href="#multiplication-table">multiplication table</a>.</p>
<h3 id="class-equation">Class equation</h3>
<p>The elements of a group can be partitioned into <a href="#conjugacy-classes">conjugacy
classes</a>. The class equation is a numerical equation
describing this partitioning. For instance, the group
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a> has three conjugacy
classes, of sizes 1, 2, and 3 respectively. The <a href="#order-of-a-group">order</a> of the group
is 6, and so the class equation is <script type="math/tex; mode=display">1 + 2 + 3 = 6.</script>
</p>
<p>All class equations are of this form: The left hand side is a sum of
positive integers, each the size of a conjugacy class, and the right side
the order of the group.</p>
<p>CITE(VGT-7.5 MM-3.7 TJ-14.2)</p>
<h3 id="closure-of-a-subset">Closure of a subset</h3>
<p>Not all subsets of a group are <a href="#subgroup">subgroups</a>. The closure of a
subset is the smallest subgroup containing that subset. That is, it answers
the question, &ldquo;What must I add to this subset to get a subgroup?&rdquo; Another
way to think of a subset&rsquo;s closure is that it is the subgroup for which the
subset is a set of <a href="#generators-for-a-group-or-subgroup">generators</a>.</p>
<h3 id="commutative-group">Commutative group</h3>
<p>See <a href="#abelian-group">Abelian</a>.</p>
<h3 id="conjugacy-classes">Conjugacy classes</h3>
<p>The conjugacy class of an element <script type="math/tex">g</script> in a group is the set of all elements <script type="math/tex">hgh^{-1}</script> for any element <script type="math/tex">h</script> in the group. These are called &ldquo;classes&rdquo; because they partition the group (that is, they form an equivalence relation) and are called &ldquo;conjugacy&rdquo; classes because the element <script type="math/tex">hgh^{-1}</script> is called the &ldquo;conjugate&rdquo; of <script type="math/tex">g</script> by <script type="math/tex">h</script>.</p>
<p>See also <a href="#class-equation">class equation</a>.</p>
<p>CITE(VGT-7.5 MM-3.7)</p>
<h3 id="cosets">Cosets</h3>
<p>For any subgroup <script type="math/tex">H</script> of a group <script type="math/tex">G</script>, we can speak of the left cosets and
the right cosets of <script type="math/tex">H</script>. The notation <script type="math/tex">aH</script> for an element <script type="math/tex">a</script> in <script type="math/tex">G</script>
means the set of all products <script type="math/tex">a</script> times an element of <script type="math/tex">H</script>; this set is a
left coset of <script type="math/tex">H</script>, because we multiplied by <script type="math/tex">a</script> on the left of <script type="math/tex">H</script>.
One could do the same on the right and form a right coset of <script type="math/tex">H</script>. The
collection of all <script type="math/tex">aH</script> for every <script type="math/tex">a</script> in <script type="math/tex">G</script> is the collection of left
cosets of <script type="math/tex">H</script>; similarly for right cosets. Cosets of either type partition
the elements of the group.</p>
<p>CITE(VGT-6.4 MM-3.2 DE-7.1 TJ-6.1)</p>
<h3 id="cycle-graph">Cycle graph</h3>
<p>A cycle graph is an illustration of the cycles of a group (<a href="#orbit-of-an-element-in-a-group">orbits of
elements</a>) and how those cycles connect.
Here is an example cycle graph.</p>
<p><img alt="Cycle graph of Z_2 times Z_4" src="../z_2_x_z_4_cycle_miniature.png" /></p>
<p>The above graph shows the group
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/Z_2
x Z_4.group"><script type="math/tex">\mathbb{Z}_2\times\mathbb{Z}_4</script></a>. One can see that there are two four-cycles (in the top half
of the picture) which share two nodes (the central node and the topmost
node). In addition to these six elements, there are two other
<a href="#order-of-an-element-in-a-group">order</a>-2 elements that are not in either
of the two larger four-cycles, shown at the bottom of the picture.</p>
<p><em>Group Explorer</em> has a <a href="../rf-geterms/#visualizers">visualizer</a> for showing
you Cycle graphs, documented in full <a href="../rf-um-cg-options/">here</a>.</p>
<p>CITE(VGT-5.1)</p>
<h3 id="cyclic-group">Cyclic group</h3>
<p>A cyclic group is one that is
<a href="#generators-for-a-group-or-subgroup">generated</a> by one element. Therefore
it is comprised entirely of the <a href="#orbit-of-an-element-in-a-group">orbit</a> of
that element. The Cyclic groups are denoted <script type="math/tex">\mathbb{Z}_n</script> or sometimes
<script type="math/tex">C_n</script>, and look just like their name (cycles) when viewed as <a href="#cayley-diagrams">Cayley
diagrams</a> or <a href="#cycle-graph">cycle graphs</a>.</p>
<p>CITE(VGT-5.1 MM-2.1 DE-6.1 TJ-4)</p>
<h3 id="definition-of-a-group-via-generators-and-relations">Definition of a group via generators and relations</h3>
<p>One can define groups by listing
<a href="#generators-for-a-group-or-subgroup">generators</a> <script type="math/tex">a,b,c,\ldots</script> and then
writing equations that describe how they relate. For instance, the <a href="#cyclic-group">cyclic
group</a> with five elements can be described as the group
generated by <script type="math/tex">a</script> where <script type="math/tex">a^5</script> is the identity element. This is written
    <script type="math/tex; mode=display">\left\langle a : a^5=e\right\rangle.</script>
The portion to the left of the colon (:) is the lone generator <script type="math/tex">a</script> and the
portion to the right is the equation that describes it.</p>
<p>For groups with more than one generator, the situation is slightly more
complex, but is essentially the same. The definition
    <script type="math/tex; mode=display">\left\langle a,b : a^4=e, b^2=e, ab=ba\right\rangle</script>
describes a group with an <a href="#order-of-an-element-in-a-group">order</a>-4
generator <script type="math/tex">a</script> and an <a href="#order-of-an-element-in-a-group">order</a>-2 generator
<script type="math/tex">b</script> which commute with each other. The last equation <script type="math/tex">ab=ba</script> describes
the commutativity of the generators, and therefore implies the commutativity
of the whole group. This group is thus
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/Z_2
x Z_4.group"><script type="math/tex">\mathbb{Z}_2\times\mathbb{Z}_4</script></a>. But if we had written the same definition with a different
final equation, say <script type="math/tex">bab=a^{-1}</script>, we would have come up with a different
group
(<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/D_4.group"><script type="math/tex">D_4</script></a>).</p>
<p>One of the columns available for view in <a href="../rf-um-mainwindow/">the main page (the group
library)</a> is the definition in this format of every
group loaded.</p>
<p>CITE(MM-1.4)</p>
<h3 id="elementwise-product-of-two-subsets-of-a-group">Elementwise product (of two subsets of a group)</h3>
<p>The elementwise product of two subsets <script type="math/tex">S</script> and <script type="math/tex">T</script> of a group is written
<script type="math/tex">ST</script> and is the set of all products of elements from <script type="math/tex">S</script> with elements
from <script type="math/tex">T</script>, in that order. Thus the product is taken on the level of
elements, or &ldquo;elementwise.&rdquo; In set theory notation,
    <script type="math/tex; mode=display">ST=\left\{ st : s\text{ is from }S\text{ and }t\text{ is from }T \right\}.</script>
</p>
<h3 id="epimorphism">Epimorphism</h3>
<p>See <a href="#surjective-surjection">surjection</a>.</p>
<h3 id="first-isomorphism-theorem">First Isomorphism Theorem</h3>
<p>The First Isomorphism Theorem says that given a
<a href="#homomorphism">homomorphism</a> <script type="math/tex">f</script> mapping a group <script type="math/tex">G</script> to a group <script type="math/tex">G'</script>,
the <a href="#kernel-of-a-homomorphism">kernel</a> of <script type="math/tex">f</script> is a <a href="#normal-subgroup">normal
subgroup</a> of <script type="math/tex">G</script>, and when we take the
<a href="#quotient-group">quotient</a> <script type="math/tex">\frac{G}{\text{Ker }f}</script>, it is
<a href="#isomorphism-isomorphic">isomorphic</a> to the image <script type="math/tex">\text{Im}(f)</script> as a
subgroup of <script type="math/tex">G'</script>.</p>
<p>This is useful in <a href="#short-exact-sequence">short exact sequences</a>, which
<em>Group Explorer</em> uses to illustrate the normality of subgroups. (To see an
example, open a <a href="../rf-um-groupwindow/">group info page</a>, click the &ldquo;tell me
more&rdquo; link next to the Subgroups computation, and search for &ldquo;short exact
sequence&rdquo; on that page&ndash;it will be in the description of any normal
subgroup.)</p>
<p>CITE(VGT-8.2 MM-4 DE-9 TJ-11.2)</p>
<h3 id="generators-for-a-group-or-subgroup">Generators for a group (or subgroup)</h3>
<p>A collection <script type="math/tex">C</script> of elements of a <a href="#group">group</a> is said to generate the
group (and they are called generators) if all possible combinations of
multiplications of those elements with one another yields all elements of
the group.</p>
<p>For instance, the elements <script type="math/tex">r</script> and <script type="math/tex">f</script> in
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a>
generate the group because the complete list of elements of
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a>
is <script type="math/tex">r,f,rf,fr,r^2</script>, plus the identity, which can be written as <script type="math/tex">r^3</script>.
Thus all elements of <script type="math/tex">S_3</script> are expressible as products of <script type="math/tex">r</script>s and
<script type="math/tex">f</script>s, so the set <script type="math/tex">\{r,f\}</script> generates <script type="math/tex">S_3</script>. Sometimes this is written
<script type="math/tex">\left\langle r,f \right\rangle=</script>
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a>.</p>
<p>If we consider just the element <script type="math/tex">r</script> in the same group and ask what set of
elements it generates, <script type="math/tex">\langle r\rangle</script>, we find only the elements
<script type="math/tex">\{e,r,r^2\}</script>. Therefore <script type="math/tex">r</script> does not generate all of
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a>,
but only a <a href="#subgroup">subgroup</a>.</p>
<p>CITE(VGT-1.4 VGT-2.3)</p>
<h3 id="group">Group</h3>
<p>A group is a set of elements together with a binary operation (which I&rsquo;ll denote here by <script type="math/tex">*</script>) such that the following criteria hold.</p>
<ul>
<li>The binary operation is associative:
   For every <script type="math/tex">a</script>, <script type="math/tex">b</script>, and <script type="math/tex">c</script> in the set, <script type="math/tex">a*(b*c)=(a*b)*c</script>.</li>
<li>There is a special element in the set called the identity,
   which I will here denote <script type="math/tex">e</script>.
   The identity obeys this rule: for every <script type="math/tex">a</script> in the group,
   <script type="math/tex">a*e=e*a=a</script>.</li>
<li>For every element <script type="math/tex">a</script> in the set,
   there is an element that is its inverse,
   usually written <script type="math/tex">a^{-1}</script>, such that <script type="math/tex">a*a^{-1} = e</script>.</li>
</ul>
<p>But this is the formal definition of a group. <a href="../gs-index/">You should be looking at
pictures of them!</a> That&rsquo;s what this program is for!</p>
<p>CITE(VGT-1 VGT-4 MM-1.1 DE-2 DE-5.2 TJ-3)</p>
<h3 id="homomorphism">Homomorphism</h3>
<p>Sometimes called simply a <a href="#morphism">morphism</a>, this is a function from one
group to another&ndash;that is, from the set of elements of the one group to the
set of elements of the other&ndash;that also <em>preserves</em> the group operation.
That is, it must keep the group&rsquo;s structure intact in the following specific
way: For every pair of elements <script type="math/tex">a</script> and <script type="math/tex">b</script> in the domain, the
homomorphism <script type="math/tex">f</script> must satisfy the equation <script type="math/tex">f(ab) = f(a)f(b)</script>.
&ldquo;Homomorphism&rdquo; means &ldquo;same shape/form&rdquo; (homo = same, morph = shape/form).</p>
<p>CITE(VGT-8 MM-4 DE-9 TJ-9 TJ-11)</p>
<h3 id="image-of-a-subset-under-a-morphism">Image (of a subset under a morphism)</h3>
<p>If a <a href="#homomorphism">homomorphism</a> <script type="math/tex">f</script> maps group <script type="math/tex">G</script> to group <script type="math/tex">G'</script>,
and there is a subset <script type="math/tex">S</script> of <script type="math/tex">G</script>, we can find out what its &ldquo;image&rdquo; is
under the homomorphism <script type="math/tex">f</script> by simply applying <script type="math/tex">f</script> to each element of
<script type="math/tex">S</script>. That is, the image of <script type="math/tex">S</script> under <script type="math/tex">f</script>, sometimes written <script type="math/tex">f[S]</script>,
is the set <script type="math/tex">\left\{ f(a) : a\text{ is in }S \right\}</script>. Obviously this will
be a subset of <script type="math/tex">G'</script>.</p>
<p>One can also speak of the image of <script type="math/tex">f</script>, meaning the image of the whole
group <script type="math/tex">G</script> under <script type="math/tex">f</script>, i.e. all elements of the codomain of <script type="math/tex">f</script>.</p>
<p>See also <a href="#preimage-of-a-subset-under-a-morphism">preimage</a>.</p>
<h3 id="index-of-a-subgroup">Index of a subgroup</h3>
<p>The <a href="#order-of-a-subgroup">order of a subgroup</a> will always divide the
<a href="#order-of-a-group">order of the group</a>. So we say that a subgroup&rsquo;s index
is the integer resulting from that division. In symbols, if <script type="math/tex">H</script> is a
<a href="#subgroup">subgroup</a> of <script type="math/tex">G</script>, then because <script type="math/tex">|H|</script> divides <script type="math/tex">|G|</script>, the
index is <script type="math/tex">\frac{|G|}{|H|}</script>, often written <script type="math/tex">[G:H]</script>.</p>
<p>CITE(VGT-6.5 MM-3.2 DE-7.2 TJ-6.2)</p>
<h3 id="injective-injection">Injective, injection</h3>
<p>A function is injective (or <a href="#1-1-one-to-one">1-1</a>, or an injection) if no
two different elements map to the same output. That is, if <script type="math/tex">a</script> and <script type="math/tex">b</script>
are not equal, then <script type="math/tex">f(a)</script> and <script type="math/tex">f(b)</script> are not equal either. An injective
<a href="#homomorphism">homomorphism</a> is sometimes called a
<a href="#monomorphism">monomorphism</a>.</p>
<h3 id="isomorphism-isomorphic">Isomorphism, isomorphic</h3>
<p>A <a href="#homomorphism">homomorphism</a> that is <a href="#bijection-bijective">bijective</a> is
called an isomorphism. In group theory, if there is an isomorphism from one
<a href="#group">group</a> to another, that means that those groups are really the same
exact structure, but possibly with different names or labels. In other
words, they are the same mathematical object.</p>
<p>Oftentimes people do not consider the distinction of labeling significant,
and will therefore call two objects the same if they are isomorphic. This is
what is meant by the phrase &ldquo;the same <em>up to isomorphism</em>&rdquo;; it means the
objects are the same if we consider isomorphic things to be identical.</p>
<p>CITE(VGT-8 MM-4 DE-9 TJ-9)</p>
<h3 id="the-isomorphism-theorems">The isomorphism theorems</h3>
<p>The <a href="#first-isomorphism-theorem">First Isomorphism Theorem</a> can be
illustrated in <em>Group Explorer.</em> Other isomorphism theorems do not yet
appear illustrated in this software.</p>
<h3 id="kernel-of-a-homomorphism">Kernel (of a homomorphism)</h3>
<p>The kernel of a <a href="#homomorphism">homomorphism</a> is the set of elements that
the homomorphism maps to the identity element. Sometimes written
<script type="math/tex">\text{Ker}(f)</script>, this set is the
<a href="#preimage-of-a-subset-under-a-morphism">preimage</a> of the set <script type="math/tex">\{ e \}</script>
(where <script type="math/tex">e</script> is the identity of the group to which the homomorphism maps),
in symbols <script type="math/tex">\{ a : f(a) = e \}</script>.</p>
<p>CITE(VGT-8.2 MM-4 DE-9 TJ-11.2)</p>
<h3 id="lattice-of-subgroups">Lattice of subgroups</h3>
<p>I will not define the concept of a lattice here; it is too broad a subject
for the small need we have of it. One can find a good definition of it in a
discrete mathematics textbook or
<a href="https://en.wikipedia.org/wiki/Lattice_(order)">online</a>.</p>
<p>For our purposes, a lattice is a two-dimensional arrangement of sets, with
larger objects higher in the arrangement (vertically), and with arrows drawn
from smaller objects up to larger ones if the smaller object is a subset of
the larger. Because all subgroups of a group are sets, we can arrange them
in a lattice. For example, <a href="../s_3_multtable_lattice.png">click here to see the lattice of subgroups of
<script type="math/tex">S_3</script></a> (illustrated using <a href="#multiplication-table">multiplication
tables</a>).</p>
<h3 id="left-cosets">Left cosets</h3>
<p>See <a href="#cosets">cosets</a>.</p>
<h3 id="monomorphism">Monomorphism</h3>
<p>See <a href="#injective-injection">injection</a>.</p>
<h3 id="morphism">Morphism</h3>
<p>See <a href="#homomorphism">homomorphism</a>.</p>
<h3 id="multiplication-table">Multiplication table</h3>
<p>A multiplication table for a group is so named because it is much like
elementary school multiplication tables, except that it uses the group
elements and operation rather than integers under ordinary, everyday
multiplication.</p>
<p>Thus the table is a grid, and across the top row and down the left column
every element of the group is listed, and filling the rest of the table are
the results of applying the group operation to the elements in the header
row and column. For this reason, multiplication tables very well exhibit
patterns inherent in the group <em>operation</em>, but elements themselves appear
several times in the table, and thus the group as a set is not well
depicted.</p>
<p><em>Group Explorer</em> has a <a href="../rf-geterms/#visualizers">visualizer</a> for showing
you multiplication tables. More information about it appears in its
<a href="../rf-um-mt-options/">documentation</a>, <a href="../gs-mt-intro/">introduction</a>, and
<a href="../tu-mt-manip/">tutorial</a>.</p>
<p><em>Group Explorer</em> shows multiplication tables in one of two ways&ndash;with or
without text in the cells of the table. Consider the following
multiplication table for the group
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/D_4.group"><script type="math/tex">D_4</script></a>.</p>
<p><img alt="Unlabeled multiplication table for D_4" src="../d_4_multtable_miniature_unlabelled.png" /></p>
<p>This table has no text and thus the colors of the cells exhibit the abstract
pattern inherent in the group operation. Omitting the text allows tables to
be shown in small sizes (like in <a href="../rf-um-groupwindow/">group info pages</a> or
<a href="../rf-um-mainwindow/">the group library</a>). If you want to see this same
table with the element names in the cells, <a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/D_4.group">click here to open the group
info page for
<script type="math/tex">D_4</script></a>
and then click the multiplication table visualizer shown under the Views
section. (Or jump to it directly with <a href="http://nathancarter.github.io/group-explorer/Multtable.html?groupURL=groups/D_4.group">this
link</a>.)</p>
<p>Upon inspecting the multiplication table with text included, you can see
that for example the element <script type="math/tex">r</script> in the leftmost column, second row,
multiplied by the element <script type="math/tex">f</script> (in the topmost row, fifth column) results
in the element <script type="math/tex">rf</script>, in the second row, fifth column. This is because
multiplication is done in the order you can infer from this example: element
in left column times element in top row.</p>
<p>CITE(VGT-4.3 MM-1.5 DE-5.3)</p>
<h3 id="normal-subgroup">Normal subgroup</h3>
<p>A <a href="#subgroup">subgroup</a> is called normal when any one of the following
equivalent criteria are met.</p>
<ul>
<li>A normal subgroup is one whose collection of
   <a href="#left-cosets">left cosets</a> is the same as its collection of
   <a href="#right-cosets">right cosets</a>.</li>
<li>A normal subgroup is one which is self-conjugate,
   that is if <script type="math/tex">N</script> is the subgroup then
   for any element <script type="math/tex">G</script> in the group,
   the set <script type="math/tex">gNg^{-1}=N</script>.
   Here <script type="math/tex">gNg^{-1}=\{ gng^{-1} : n\text{ is in }N\}</script>.</li>
<li>A normal subgroup <script type="math/tex">N</script> in a group <script type="math/tex">G</script> is one for which
   the <a href="#quotient-group">quotient</a> <script type="math/tex">\frac{G}{N}</script> is defined.</li>
</ul>
<p>The last of these is probably the easiest to visualize. <a href="#multiplication-table">Multiplication
tables</a> and <a href="#cayley-diagrams">Cayley diagrams</a> can
both organize themselves by the <a href="#cosets">cosets</a> of a subgroup and then
separate those cosets (or &ldquo;chunk&rdquo; them) to help you visualize the quotient
operation. Refer to the documentation on <a href="../rf-um-mt-options/">the multiplication table
interface</a> or <a href="../rf-um-cd-options/">the Cayley diagram
interface</a> for more information on these features.</p>
<p>CITE(VGT-7.3 MM-3.3 DE-7.3 TJ-10)</p>
<h3 id="normalizer-of-a-subgroupsubset">Normalizer of a subgroup/subset</h3>
<p>The normalizer of a <a href="#subgroup">subgroup</a> <script type="math/tex">H</script> of a group <script type="math/tex">G</script>, sometimes
written <script type="math/tex">\text{Norm}(H)</script>, is the largest subgroup containing <script type="math/tex">H</script> in
which <script type="math/tex">H</script> is <a href="#normal-subgroup">normal</a>.</p>
<p>That is, <script type="math/tex">H</script> may not be normal in <script type="math/tex">G</script>, but if we were to remove some of
the &ldquo;problem&rdquo; elements from <script type="math/tex">G</script>, those that are preventing <script type="math/tex">H</script> from
being normal, we would find a subgroup of <script type="math/tex">G</script> in which <script type="math/tex">H</script> is normal.
The normalizer is exactly this, the subgroup which remains when you remove
as few elements as possible from <script type="math/tex">G</script> to make <script type="math/tex">H</script> normal.</p>
<p>CITE(VGT-7.4 MM-3.6)</p>
<h3 id="objects-of-symmetry">Objects of symmetry</h3>
<p>Group theory is the study of symmetry in the abstract. But many very
concrete objects which one could hold in one&rsquo;s hand have symmetry (or
symmetries) to them, and the relationship among those symmetries can be
described by a group. Therefore several groups in <em>Group Explorer&rsquo;s</em> library
are groups that describe the symmetry of objects that exist in
three-dimensional space, and exist in the real world&ndash;for example, a cube, a
pinwheel, or a pyramid. These objects are called &ldquo;objects of symmetry&rdquo; (or
objects <em>with</em> symmetry, or symmetry objects).</p>
<p><em>Group Explorer</em> has a <a href="../rf-geterms/#visualizers">visualizer</a> for showing
you objects of symmetry, documented in full <a href="../rf-um-os-options/">here</a>.</p>
<p>CITE(VGT-3 MM-1.3 TJ-12.2)</p>
<h3 id="one-to-one">One-to-one</h3>
<p>See <a href="#injective-injection">injective</a>.</p>
<h3 id="onto">Onto</h3>
<p>See <a href="#surjective-surjection">surjective</a>.</p>
<h3 id="orbit-of-an-element-in-a-group">Orbit of an element in a group</h3>
<p>The orbit of an element a in a <a href="#group">group</a> <script type="math/tex">G</script> is the set of all
powers of that element, i.e. <script type="math/tex">\{ a^n : n\text{ is any integer} \}</script>. You
can think of this as all the places one can &ldquo;walk&rdquo; using <script type="math/tex">a</script> as a step. In
a <a href="#cayley-diagrams">Cayley diagram</a>, this would be the chain of elements
one encounters by following <script type="math/tex">a</script>-arrows repeatedly.</p>
<p>CITE(VGT-5.1)</p>
<h3 id="order-classes">Order classes</h3>
<p>Each element in a <a href="#group">group</a> has an
<a href="#order-of-an-element-in-a-group">order</a>, and thus we can partition the
elements of the group into classes which all have the same order. For
instance the elements of the group
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/S_3.group"><script type="math/tex">S_3</script></a>
are listed in the table below, with their orders.</p>
<table>
<thead>
<tr>
<th align="center">Element</th>
<th align="center">Order</th>
</tr>
</thead>
<tbody>
<tr>
<td align="center">
<script type="math/tex">e</script>
</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">
<script type="math/tex">r</script>
</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">
<script type="math/tex">r^2</script>
</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">
<script type="math/tex">f</script>
</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">
<script type="math/tex">fr</script>
</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">
<script type="math/tex">rf</script>
</td>
<td align="center">2</td>
</tr>
</tbody>
</table>
<p>Thus this group has three order classes: one consisting of the elements of
order 1, <script type="math/tex">\{ e \}</script>, another consisting of the elements of order 2, <script type="math/tex">\{ f,
fr, rf \}</script>, and another consisting of the elements of order 3, <script type="math/tex">\{ r, r^2
\}</script>.</p>
<p>You can learn about the order classes of any group by looking under the
Computations section of its <a href="../rf-um-groupwindow/">group info page</a>.</p>
<h3 id="order-of-an-element-in-a-group">Order of an element in a group</h3>
<p>The order of an element in a <a href="#group">group</a> can be thought of in two
equivalent ways.</p>
<ul>
<li>The order of the element <script type="math/tex">a</script> is the smallest positive power of a
   that yields the identity. That is, if <script type="math/tex">a^k=e</script>
   and no smaller positive integer exponent satisfies that same equation,
   then the order of <script type="math/tex">a</script>, written <script type="math/tex">|a|</script>, is <script type="math/tex">k</script>.</li>
<li>The order of the element <script type="math/tex">a</script> is the
   <a href="#order-of-a-subgroup">order of the subgroup</a>
   <a href="#generators-for-a-group-or-subgroup">generated by</a> <script type="math/tex">a</script>,
   which is also the <a href="#orbit-of-an-element-in-a-group">orbit</a> of <script type="math/tex">a</script>.</li>
</ul>
<p>The order of the identity element is 1 by either of these reckonings.
It generates the <a href="#subgroup">subgroup</a> <script type="math/tex">\{ e \}</script>.</p>
<h3 id="order-of-a-group">Order of a group</h3>
<p>The order of a group <script type="math/tex">G</script>, written <script type="math/tex">|G|</script>, is simply its size (how many
elements are in it).</p>
<p>CITE(VGT-5.1)</p>
<h3 id="order-of-a-subgroup">Order of a subgroup</h3>
<p>The order of a subgroup <script type="math/tex">H</script>, written <script type="math/tex">|H|</script>, is simply its size (how many
elements are in it). See also <a href="#index-of-a-subgroup">subgroup index</a>.</p>
<h3 id="p45subgroup">
<script type="math/tex">p</script>-subgroup</h3>
<p>A <script type="math/tex">p</script>-subgroup is a <a href="#subgroup">subgroup</a> all of whose elements have an
<a href="#order-of-an-element-in-a-group">order</a> equal to a power of the prime
number <script type="math/tex">p</script>. (See also <a href="#sylow-p-subgroup">Sylow <script type="math/tex">p</script>-subgroup</a>.)</p>
<p>For instance, if every element of <script type="math/tex">H</script> has order <script type="math/tex">1, 2, 4, 8, 16,\ldots</script>
(any power of 2), then <script type="math/tex">H</script> is a 2-subgroup. If every element of <script type="math/tex">H</script> has
order <script type="math/tex">1, 5, 25, 125,\ldots</script> (any power of 5), then <script type="math/tex">H</script> is a 5-subgroup.
One does not say &ldquo;4-subgroup&rdquo; or &ldquo;20-subgroup&rdquo; because those numbers are not
prime. Also, one can see that if <script type="math/tex">H</script> is a <script type="math/tex">p</script>-subgroup for some prime
<script type="math/tex">p</script>, then it is not a <script type="math/tex">q</script>-subgroup for any other prime <script type="math/tex">q</script> unless
<script type="math/tex">H= \{ e \}</script>.</p>
<p>If you click &ldquo;tell me more&rdquo; next to Subgroups in the Computations section of
any <a href="../rf-um-groupwindow/">group info page</a>, you will see that the
descriptions of the subgroups tell you which ones are <script type="math/tex">p</script>-subgroups.</p>
<p>CITE(VGT-9.3 MM-5.5 TJ-15)</p>
<h3 id="preimage-of-a-subset-under-a-morphism">Preimage (of a subset under a morphism)</h3>
<p>If a <a href="#homomorphism">homomorphism</a> <script type="math/tex">f</script> maps a group <script type="math/tex">G</script> to a group
<script type="math/tex">G'</script> and there is a subset <script type="math/tex">S</script> of <script type="math/tex">G'</script>, we can find out what its
&ldquo;pre-image&rdquo; is under the homomorphism <script type="math/tex">f</script> by finding all the elements
which <script type="math/tex">f</script> maps into <script type="math/tex">S</script>. That is, the preimage of <script type="math/tex">S</script> under <script type="math/tex">f</script>,
sometimes written <script type="math/tex">f^{-1}[S]</script>, is the set <script type="math/tex">\left\{ a : f(a)\text{ is in
}S \right\}</script>. Obviously this will be a subset of <script type="math/tex">G</script>. See also
<a href="#image-of-a-subset-under-a-morphism">image</a>.</p>
<h3 id="proper-subgroup">Proper subgroup</h3>
<p>A <a href="#subgroup">subgroup</a> is proper if it is not the whole <a href="#group">group</a>.</p>
<p>Technically, by <a href="#subgroup">the definition of subgroup</a>, every group is a
subgroup of itself. But when we say &ldquo;a proper subgroup&rdquo; we mean subgroups
that are actually smaller than the group we&rsquo;re looking inside.</p>
<h3 id="quotient-group">Quotient group</h3>
<p>If <script type="math/tex">H</script> is a <a href="#subgroup">subgroup</a> of <script type="math/tex">G</script>, then we can sometimes make a
group out of the <a href="#cosets">cosets</a> of <script type="math/tex">H</script> as follows. The cosets of <script type="math/tex">H</script>
(let&rsquo;s use the <a href="#left-cosets">left cosets</a>) are the sets <script type="math/tex">aH</script> for various
values of <script type="math/tex">a</script>. To define an operation on this collection, we let <script type="math/tex">aH</script>
times <script type="math/tex">bH</script> equal <script type="math/tex">(ab)H</script>. One can prove that this operation is a valid
group operation (as per <a href="#group">the definition of a group</a>) if and only if
<script type="math/tex">H</script> is a <a href="#normal-subgroup">normal subgroup</a> of <script type="math/tex">G</script>. In that case, the
new group we just formed is called the quotient of <script type="math/tex">G</script> by <script type="math/tex">H</script>, written
<script type="math/tex">\frac{G}{H}</script>.</p>
<p><a href="#multiplication-table">Multiplication tables</a> and <a href="#cayley-diagrams">Cayley
diagrams</a> can both organize themselves by the
<a href="#cosets">cosets</a> of a subgroup and then separate those cosets (or &ldquo;chunk&rdquo;
them) to help you visualize the quotient operation. Refer to the
documentation on <a href="../rf-um-mt-options/">the multiplication table interface</a>
or <a href="../rf-um-cd-options/">the Cayley diagram interface</a> for more information
on these features.</p>
<p>CITE(VGT-7.3 MM-3.5 DE-8.2 TJ-10)</p>
<h3 id="right-cosets">Right cosets</h3>
<p>See <a href="#cosets">cosets</a>.</p>
<h3 id="short-exact-sequence">Short exact sequence</h3>
<p>An exact sequence is a chain of groups connected by
<a href="#homomorphism">homomorphisms</a> such that the
<a href="#image-of-a-subset-under-a-morphism">image</a> of any one homomorphism in the
chain is the <a href="#kernel-of-a-homomorphism">kernel</a> of the next homomorphism. A
short exact sequence is one with only five groups in it, the first and last
of which are both <a href="#trivial-groupsubgroup">the trivial group</a>. An example
is shown below.</p>
<p><img alt="A sheet showing a short exact sequence for the normality of V_4 in A_4" src="../a_4_sheet_ses.png" /></p>
<p>A short exact sequence is related to the <a href="#quotient-group">quotient</a>
operation on groups. Let us call the four morphisms in a short exact
sequence <script type="math/tex">id</script>, <script type="math/tex">e</script>, <script type="math/tex">q</script>, and <script type="math/tex">z</script>, from left to right, and the three
middle groups in the sequence <script type="math/tex">A</script>, <script type="math/tex">B</script>, and <script type="math/tex">C</script>, also from left to
right.</p>
<p>I call the first homomorphism <script type="math/tex">id</script> (for &ldquo;identity&rdquo;) because it simply maps
the one element in its domain (the identity element) to the identity element
in <script type="math/tex">A</script>.</p>
<p>As per the definition above, the kernel of <script type="math/tex">e</script> must be the image of
<script type="math/tex">id</script>, which is simply the identity element. A nice theorem of group theory
tells us that when a morphisms&rsquo; kernel is the identity, the whole morphism
is <a href="#injective-injection">injective</a>; thus a copy of <script type="math/tex">A</script> appears in <script type="math/tex">B</script>,
that copy being the image of <script type="math/tex">e</script> (which stands for &ldquo;embedding,&rdquo; a synonym
for <a href="#injective-injection">injection</a>).</p>
<p>In turn, the <a href="#kernel-of-a-homomorphism">kernel</a> of <script type="math/tex">q</script> must be the image
of <script type="math/tex">e</script>, which is an <a href="#isomorphism-isomorphic">isomorphic</a> copy of <script type="math/tex">A</script>.
This means that the map <script type="math/tex">q</script> effectively zeroes out or removes <script type="math/tex">A</script> from
<script type="math/tex">B</script>. The <a href="#first-isomorphism-theorem">First Isomorphism Theorem</a> then
implies that <script type="math/tex">q</script> is a quotient map (hence the name) and that <script type="math/tex">\frac{B}{A}
= \text{Im}(q)</script>.</p>
<p>But there&rsquo;s more! Because the image of <script type="math/tex">z</script> is only the identity element,
its <a href="#kernel-of-a-homomorphism">kernel</a> is all of <script type="math/tex">C</script>. Therefore because
this is an exact sequence, the image of <script type="math/tex">q</script> must also be <script type="math/tex">C</script>. This means
that the short exact sequence <script type="math/tex">A\to B\to C</script> illustrates the fact that
<script type="math/tex">\frac{B}{A}=C</script>.</p>
<p>You can see a short exact sequence illustrated for any <a href="#normal-subgroup">normal
subgroup</a> of any group. Go to a <a href="../rf-um-groupwindow/">group&rsquo;s info
page</a>, to the Subgroups section under Computations,
and click &ldquo;tell me more.&rdquo; Any normal subgroup will provide a link to a
<a href="../rf-geterms/#sheet">sheet</a> illustrating the <a href="#quotient-group">quotient</a>
via a short exact sequence.</p>
<h3 id="simple-group">Simple group</h3>
<p>A simple group is a non-<a href="#abelian-group">abelian group</a> with no
non-<a href="#trivial-groupsubgroup">trivial</a>, <a href="#proper-subgroup">proper</a>, <a href="#normal-subgroup">normal
subgroups</a>. The smallest simple group is
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/A_5.group"><script type="math/tex">A_5</script></a>.</p>
<p>CITE(MM-5.7 TJ-10.2)</p>
<h3 id="solvable-group-solvable-decomposition">Solvable group, solvable decomposition</h3>
<p>Solvable groups are important in Galois theory, which is too large a topic
to embark on here. Briefly, Galois theory was invented to study which
polynomials are solvable using ordinary arithmetic plus radicals. If you are
interested in Galois theory, refer to an abstract algebra textbook or <a href="https://en.wikipedia.org/wiki/Galois_theory">an
online resource</a>.</p>
<p>The roots of polynomials have symmetry that can be described by groups,
called the Galois group of the polynomials. Evariste Galois (a 19th century
mathematician) proved that you could tell by looking at these groups whether
the polynomial was solvable. Groups that corresponded to solvable
polynomials got the name &ldquo;solvable groups.&rdquo;</p>
<p>A group <script type="math/tex">G</script> is solvable if there is a chain of groups <script type="math/tex">H_1, H_2, H_3,
\ldots, H_n</script> such that each group is a <a href="#normal-subgroup">normal subgroup</a>
of the next one in the chain, the resulting <a href="#quotient-group">quotient
groups</a> are all abelian, and the chain begins with <a href="#trivial-groupsubgroup">the
trivial group</a> and ends with <script type="math/tex">G</script>. You can see a
diagram illustrating this for any solvable group by looking under the
Computations section of the <a href="../rf-um-groupwindow/">group&rsquo;s info page</a>.</p>
<p>CITE(VGT-10 MM-6 TJ-13.2)</p>
<h3 id="subgroup">Subgroup</h3>
<p>If <script type="math/tex">S</script> is a subset of the <a href="#group">group</a> <script type="math/tex">G</script> (i.e. a subset of the set
of elements of <script type="math/tex">G</script>) then we say <script type="math/tex">S</script> is a subgroup if it is also a group
under the operation of <script type="math/tex">G</script>.</p>
<p>A subset <script type="math/tex">S</script> of a group <script type="math/tex">G</script> may fail to be a subgroup in a few different ways; here are examples.</p>
<ul>
<li>If <script type="math/tex">S</script> does not contain the identity element,
   it violates one of the criteria in <a href="#group">the definition of a group</a>.</li>
<li>If <script type="math/tex">S</script> contains an element but not that element&rsquo;s inverse,
   it would violates another of those criteria.</li>
<li>If <script type="math/tex">S</script> contains two elements but not their product,
   then the binary operation of <script type="math/tex">G</script> cannot be said to be
   a binary operation on <script type="math/tex">S</script>,
   because it maps some pairs from <script type="math/tex">S</script> outside of <script type="math/tex">S</script>.</li>
</ul>
<p>CITE(VGT-6 MM-3.1 DE-4.1 TJ-3.3)</p>
<h3 id="surjective-surjection">Surjective, surjection</h3>
<p>A function is surjective (or <a href="#onto">onto</a>, or a surjection) from a set
<script type="math/tex">A</script> to a set <script type="math/tex">B</script> if every element of <script type="math/tex">B</script> is mapped to by some element
of <script type="math/tex">A</script>. That is, if <script type="math/tex">b</script> is in <script type="math/tex">B</script>, then there must be some <script type="math/tex">a</script> in
<script type="math/tex">A</script> such that <script type="math/tex">f(a) = b</script>. A surjective <a href="#homomorphism">homomorphism</a> is
sometimes called an <a href="#epimorphism">epimorphism</a>.</p>
<h3 id="sylow-p-subgroup">Sylow p-subgroup</h3>
<p>A Sylow <a href="#p45subgroup"><script type="math/tex">p</script>-subgroup</a> is a maximal <script type="math/tex">p</script>-subgroup; that
is, no subgroup properly containing this one is still a <script type="math/tex">p</script>-subgroup.</p>
<p>If you click &ldquo;tell me more&rdquo; next to Subgroups in the Computations section of
any <a href="../rf-um-groupwindow/">group info page</a>, you will see that the
descriptions of the subgroups tell you which ones are Sylow
<script type="math/tex">p</script>-subgroups.</p>
<p>CITE(VGT-9 MM-5.6 TJ-15)</p>
<h3 id="symmetry-objects">Symmetry objects</h3>
<p>See <a href="#objects-of-symmetry">objects of symmetry</a>.</p>
<h3 id="trivial-groupsubgroup">Trivial group/subgroup</h3>
<p>The trivial group is the <a href="#group">group</a> with only one element. <a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=groups/Trivial.group">You can see
its information
here.</a></p>
<p>In every group, the set containing only the identity element is a
<a href="#subgroup">subgroup</a> and is called the trivial subgroup.</p>
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